It is a critical task to keep the ventilation system working in a proper and efficient manner in large multi-storey buildings, and the enthalpy exchanger is becoming an increasingly important part of the ventilation system by playing the function of channeling heat and moisture. We present a computational study on the heat transfer performance of the cross-flow enthalpy exchanger, which is in large use for residential buildings. The ducts are considered whose cross-sectional shapes resemble triangle and longitudinal centerline a cosine wave. It is shown that, as the cross-sectional shape departs from triangle, the heat transfer performance of the duct tends to deteriorate. Also, applying the wave-like shape to the longitudinal centerline of the duct increases the rate of heat transfer and the applied pressure-gradient at the same time. The origin of the performance variations in the cases considered are quantitatively analyzed and discussed.